maximal ideals of ring of formal power series

Also the converse is true, i.e. if 𝔐 is a maximal ideal of R⁢[[X]], then there is a maximal ideal
𝔪 of R such that 𝔐=𝔪+(X).

Note. In the special case that R is a field, the only maximal ideal of which is the zero ideal(0), this corresponds to the only maximal ideal (X) of R⁢[[X]] (see http://planetmath.org/node/12087formal power series over field).

We here prove the first assertion. So, 𝔪 is assumed to be maximal. Let

f⁢(x):=a0+a1⁢X+a2⁢X2+…

be any formal power series in R⁢[[X]]∖𝔐. Hence, the constant term a0 cannot lie in 𝔪. According to the criterion for maximal ideal, there is an element r of R such that 1+r⁢a0∈𝔪. Therefore